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Properties

Label 4-672e2-1.1-c1e2-0-28
Degree $4$
Conductor $451584$
Sign $1$
Analytic cond. $28.7933$
Root an. cond. $2.31645$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·7-s + 9-s − 8·13-s + 12·19-s + 4·21-s − 10·25-s − 4·27-s + 8·31-s + 20·37-s − 16·39-s − 8·43-s + 3·49-s + 24·57-s − 16·61-s + 2·63-s + 16·67-s − 12·73-s − 20·75-s + 32·79-s − 11·81-s − 16·91-s + 16·93-s − 4·97-s + 8·103-s + 20·109-s + 40·111-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.755·7-s + 1/3·9-s − 2.21·13-s + 2.75·19-s + 0.872·21-s − 2·25-s − 0.769·27-s + 1.43·31-s + 3.28·37-s − 2.56·39-s − 1.21·43-s + 3/7·49-s + 3.17·57-s − 2.04·61-s + 0.251·63-s + 1.95·67-s − 1.40·73-s − 2.30·75-s + 3.60·79-s − 1.22·81-s − 1.67·91-s + 1.65·93-s − 0.406·97-s + 0.788·103-s + 1.91·109-s + 3.79·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(451584\)    =    \(2^{10} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(28.7933\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 451584,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.881300729\)
\(L(\frac12)\) \(\approx\) \(2.881300729\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.19.am_cw
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.37.au_gs
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.a_da
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.a_dy
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.59.a_s
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.79.abg_py
83$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.83.a_gg
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.a_afq
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.247827016681358338699106327510, −8.142737265603496815088661734936, −7.64230674264070832489473958476, −7.53632864653844024945923131798, −7.03619200467257105345639081242, −6.19321040018157050389604631782, −5.72643285643591914021095909045, −5.20704742703837539203927825545, −4.62285613804853643208241606946, −4.38921941756740989472338293857, −3.39651452170084757716747861999, −3.09009811816266066443914661488, −2.38160133368819946421648201021, −2.00516068697428410265218115685, −0.867770609357001154731944493797, 0.867770609357001154731944493797, 2.00516068697428410265218115685, 2.38160133368819946421648201021, 3.09009811816266066443914661488, 3.39651452170084757716747861999, 4.38921941756740989472338293857, 4.62285613804853643208241606946, 5.20704742703837539203927825545, 5.72643285643591914021095909045, 6.19321040018157050389604631782, 7.03619200467257105345639081242, 7.53632864653844024945923131798, 7.64230674264070832489473958476, 8.142737265603496815088661734936, 8.247827016681358338699106327510

Graph of the $Z$-function along the critical line